An integer $m$ is Fortunate if it can be written as $q-P$, where $P$ is a primorial and $q$ is the smallest prime greater than $P+1$. It is conjectured that Fortunate numbers are always prime.

It is easy to see that there are only finitely many possible primorials $P$ for which a given $m$ can be decomposed in the above manner (this is because $m$ must be greater than the largest prime dividing $P$).

QUESTION: Is the number of such representations of an integer $m$ uniformly bounded above?

I think you probably meant to say "smallest prime greater than $P+1$." For anyone who, like me two minutes ago, doesn't know what a primorial is: that's a portmanteau of "prime" and "factorial", i.e., multiply the first n primes together to get the $n^{th}$ primorial.
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Todd Trimble♦Sep 5 '12 at 3:23

Thanks for the correction; I fixed the question so that it doesn't contradict Euclid.
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Jon CohenSep 6 '12 at 15:36

1

The Fortunate numbers are tabulated at oeis.org/A005235. The numbers are actually named after R F Fortune, who is credited with the conjecture that they are all prime. A number of references are given at the oeis page. 23 is the smallest "doubly Fortunate" number; 61, the smallest triply Fortunate. 2 and 11 are the smallest unFortunate primes.
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Gerry MyersonSep 6 '12 at 23:58